Introduction
Main result and motivation
Corollaries and outlook
Proof

Asymptotic expansions for fractional stochastic volatility models
Blanka Horvath
Department of Mathematics, Imperial College London

3rd London-Paris Bachelier Workshop on Mathematical Finance
Based on joint works with
with Philipp Harms and Antoine Jacquier and with
Christian Bayer, Peter Friz, Archil Gulisashvili and Benjamin Stemper
September 29, 2016

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Implied volatility

Implied volatility• Asset price process: (St = eXt )t≥0 , with X0 = 0.
• No dividend, no interest rate.
• Black-Scholes-Merton (BSM) framework:= N (d+ ) − ek N (d− ) ,CBS (τ, k, σ) := E0 eXτ − ek+k1 √d± := − √ ± σ τ .
σ τ2• Spot implied volatility στ (k): the unique (non-negative) solution to
Cobserved (τ, k) = CBS (τ, k, στ (k)).
• Implied volatility: unit-free measure of option prices.

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Implied volatility

Implied volatility• Asset price process: (St = eXt )t≥0 , with X0 = 0.
• No dividend, no interest rate.
• Black-Scholes-Merton (BSM) framework:= N (d+ ) − ek N (d− ) ,CBS (τ, k, σ) := E0 eXτ − ek+k1 √d± := − √ ± σ τ .
σ τ2• Spot implied volatility στ (k): the unique (non-negative) solution to
Cobserved (τ, k) = CBS (τ, k, στ (k)).
• Implied volatility: unit-free measure of option prices.

However the implied volatility is not available in closed form for most models. Its
asymptotic behaviour is available via (small/large k, τ ) approximations.

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Implied volatility

Implied volatility (στ (k)) asymptotics as |k| ↑ ∞, τ ↓ 0 or τ ↑ ∞:• Hagan-Kumar-Lesniewski-Woodward (2003/2015): small-maturity for the SABR

model
• Berestycki-Busca-Florent (2004): small-τ using PDE methods for diffusions.
• Henry-Labordère (2009): small-τ asymptotics using differential geometry.
• Forde et al.(2012), Jacquier et al.(2012): small/ large τ using large deviations.
• Lee (2003), Benaim-Friz (2009), Gulisashvili (2010-2012), Caravenna-Corbetta(2016), De Marco-Jacquier-Hillairet (2013): |k| ↑ ∞.
• Laurence-Gatheral-Hsu-Ouyang-Wang (2012): small-τ in local volatility models.
• Fouque et al.(2000-2011): perturbation techniques for slow and fastmean-reverting stochastic volatility models.
• Mijatović-Tankov (2012): small-τ for jump models.
• Bompis-Gobet (2015): asymptotic expansions in the presence of both local andstochastic volatility using Malliavin calculus.
Related works:• Deuschel-Friz-Jacquier-Violante (CPAM 2014), De Marco-Friz (2014): small-noise

expansions using Laplace method on Wiener space (Ben Arous-Bismut approach).
• Baudoin-Ouyang (2010): small-noise expansions in a fractional setting• Forde-Zhang (2015): large deviations in a fractional stochastic volatility setting• Guennon-Jacquier-Roome (2015): large deviations in a fractional Heston model
Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Implied volatility

Motivation• Classical stochastic volatility models generate a constant short-maturity ATM

skew and a large-maturity one proportional to τ −1 ;• However, short-term data suggests a time decay of the ATM skew proportional toτ −α , α ∈ (0, 1/2).
• One solution: adding volatility factors (risk of over-parameterisation).

Gatheral’s Double Mean-Reverting, Bergomi-Guyon, each factor acting on a
specific time horizon.
• In the Lévy case (Tankov, 2010), the situation is different, as τ ↓ 0:R
• in the pure jump case with|x|ν(dx) < ∞, then στ2 (0) ∼ cτ ;(−1,1)
2• in the (α) stable case, στ(0) ∼ cτ 1−2/α for α ∈ (1, 2);2• for out-of-the-money options, στ (k) ∼Blanka Horvathk2.
2τ | log(τ )|Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Implied volatility

Rough volatility models• Gatheral-Jaisson-Rosenbaum and Bayer-Gatheral-Friz (2014,1015) proposed afractional volatility model:dSt
σt= St (σt dZt + µt dt),
= exp (Xt ) ,

(1)

where
Xt = µWtH − α(Xt − m)dt,
for µ, α > 0, m ∈ R for a Bm Z and a fBm motion W H with Hurst parameter H.
• Time series of the Oxford-Man SPX realised variance as well as implied volatility

smiles of the SPX suggest that H ∈ (0, 1/2): short-memory volatility.
• Is not statistically rejected by Ait-Sahalia-Jacod’s test (2009) for Itô diffusions.
• Main drawback: loss of Markovianity (H 6= 1/2) rules out PDE techniques, andMonte Carlo is computationally intensive.

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Implied volatility

Today’s menu
1 Introduction

Implied volatility
2 Main result and motivation

Our framework and its scope
ExamplesDensity asymptotics
3 Corollaries and outlook

Short-time expansion
Tail expansion
Implied volatility asympotics
Outlook: Refined expansions and moderate regimes
4 ProofNotations
Sketch of the proof

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Our framework and its scope
ExamplesDensity asymptotics

General setting:dXt = b1 (κ1 , Xt , Yt )dt + β σ11 (Xt , Yt )dWtH1 + σ12 (Xt , Yt )dWtH2

dYt = b2 (κ2 , Xt , Yt )dt + β σ21 (Xt , Yt )dWtH1 + σ22 (Xt , Yt )dWtH2 .

Blanka Horvath(2)

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Our framework and its scope
ExamplesDensity asymptotics

General setting:dXt = b1 (κ1 , Xt , Yt )dt + β σ11 (Xt , Yt )dWtH1 + σ12 (Xt , Yt )dWtH2

dYt = b2 (κ2 , Xt , Yt )dt + β σ21 (Xt , Yt )dWtH1 + σ22 (Xt , Yt )dWtH2 .

(2)

Motivation: ”Classical” (Hi = 12 ) case: Deuschel-Friz-Jacquier-Violante (2011)
dXt = b(, Xt )dt + mXσi (Xt )dWti ,X0 = x0 ∈ Rd

i=1

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Our framework and its scope
ExamplesDensity asymptotics

General setting:dXt = b1 (κ1 , Xt , Yt )dt + β σ11 (Xt , Yt )dWtH1 + σ12 (Xt , Yt )dWtH2

dYt = b2 (κ2 , Xt , Yt )dt + β σ21 (Xt , Yt )dWtH1 + σ22 (Xt , Yt )dWtH2 .

(2)

Motivation: ”Classical” (Hi = 12 ) case: Deuschel-Friz-Jacquier-Violante (2011)
dXt = b(, Xt )dt + mXσi (Xt )dWti ,X0 = x0 ∈ Rd

i=1

Fractional (Hi = H ∈ ( 14 , 1)) case: Baudoin-Ouyang (2015)
dXt = b(, Xt )dt + mXσi (Xt )d(W H )it ,X0 = x0 ∈ Rd

i=1

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Our framework and its scope
ExamplesDensity asymptotics

General setting:dXt = b1 (κ1 , Xt , Yt )dt + β σ11 (Xt , Yt )dWtH1 + σ12 (Xt , Yt )dWtH2

dYt = b2 (κ2 , Xt , Yt )dt + β σ21 (Xt , Yt )dWtH1 + σ22 (Xt , Yt )dWtH2 .

(2)

Motivation: ”Classical” (Hi = 12 ) case: Deuschel-Friz-Jacquier-Violante (2011)
dXt = b(, Xt )dt + mXσi (Xt )dWti ,X0 = x0 ∈ Rd

i=1

Fractional (Hi = H ∈ ( 14 , 1)) case: Baudoin-Ouyang (2015)
dXt = b(, Xt )dt + mXσi (Xt )d(W H )it ,X0 = x0 ∈ Rd

i=1

Assumptions made: b(ε, ·) → σ0 (·) and x0ε → x0 as ε → 0, the weak Hörmander
condition for {σ0 , σ1 , . . . , σd } at x0 , and vector fields are C ∞ -bounded (can berelaxed).

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Our framework and its scope
ExamplesDensity asymptotics

General setting:dXt = b1 (κ1 , Xt , Yt )dt + β σ11 (Xt , Yt )dWtH1 + σ12 (Xt , Yt )dWtH2

dYt = b2 (κ2 , Xt , Yt )dt + β σ21 (Xt , Yt )dWtH1 + σ22 (Xt , Yt )dWtH2 .

(2)

Motivation: ”Classical” (Hi = 12 ) case: Deuschel-Friz-Jacquier-Violante (2011)
dXt = b(, Xt )dt + mXσi (Xt )dWti ,X0 = x0 ∈ Rd

i=1

Fractional (Hi = H ∈ ( 14 , 1)) case: Baudoin-Ouyang (2015)
dXt = b(, Xt )dt + mXσi (Xt )d(W H )it ,X0 = x0 ∈ Rd

i=1

Assumptions made: b(ε, ·) → σ0 (·) and x0ε → x0 as ε → 0, the weak Hörmander
condition for {σ0 , σ1 , . . . , σd } at x0 , and vector fields are C ∞ -bounded (can berelaxed).
Our main interest: H1 = 21 , H2 6= 12 .
Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Our framework and its scope
ExamplesDensity asymptotics

Our setting:dXt = b1 (κ1 , Yt )dt + β σ1 (Yt )dWt
dYt = b2 (κ2 , Yt )dt + β σ2 dWtH .

Blanka Horvath(2)

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Our framework and its scope
ExamplesDensity asymptotics

Our setting:dXt = b1 (κ1 , Yt )dt + β σ1 (Yt )dWt
dYt = b2 (κ2 , Yt )dt + β σ2 dWtH .

(2)

σ1 α-Hölder continuous, α ∈ (0, 1], σ2 > 0, (but can be extended to σ2 (·) boundedand elliptic) conditions on b1 , b2 dictated by scaling and existence, and H 6= 1/2,
particular interest in H < 1/2.

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Our framework and its scope
ExamplesDensity asymptotics

Our setting:dXt = b1 (κ1 , Yt )dt + β σ1 (Yt )dWt
dYt = b2 (κ2 , Yt )dt + β σ2 dWtH .

(2)

σ1 α-Hölder continuous, α ∈ (0, 1], σ2 > 0, (but can be extended to σ2 (·) boundedand elliptic) conditions on b1 , b2 dictated by scaling and existence, and H 6= 1/2,
particular interest in H < 1/2.
To introduce correlation we consider B̃ and B independent, and set W = ρ̄B̃ + ρB and
Z t
K (t, s)Bs
WH =0

where K the Volterra kernel of the (standard) fBm W H .

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Our framework and its scope
ExamplesDensity asymptotics

Our setting:dXt = b1 (κ1 , Yt )dt + β σ1 (Yt )dWt
dYt = b2 (κ2 , Yt )dt + β σ2 dWtH .

(2)

σ1 α-Hölder continuous, α ∈ (0, 1], σ2 > 0, (but can be extended to σ2 (·) boundedand elliptic) conditions on b1 , b2 dictated by scaling and existence, and H 6= 1/2,
particular interest in H < 1/2.
To introduce correlation we consider B̃ and B independent, and set W = ρ̄B̃ + ρB and
Z t
K (t, s)Bs
WH =0

where K the Volterra kernel of the (standard) fBm W H .
Remark: The setting of Forde-Zhang ’16 is included. With a leap of faithGatheral-Jaisson-Rosenbaum ’14 and Bayer-Friz-Gatheral ’15 as well.

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Our framework and its scope
ExamplesDensity asymptotics

Our setting:dXt = b1 (κ1 , Yt )dt + β σ1 (Yt )dWt ,X0 = x0

dYt = b2 (κ2 , Yt )dt + β σ2 dWtH ,Y0 = y0 .

Blanka Horvath(2)

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Our framework and its scope
ExamplesDensity asymptotics

Our setting:dXt = b1 (κ1 , Yt )dt + β σ1 (Yt )dWt ,X0 = x0

dYt = b2 (κ2 , Yt )dt + β σ2 dWtH ,Y0 = y0 .

Blanka Horvath(2)

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Our framework and its scope
ExamplesDensity asymptotics

Our setting:dXt = b1 (κ1 , Yt )dt + β σ1 (Yt )dWt ,X0 = x0

dYt = b2 (κ2 , Yt )dt + β σ2 dWtH ,Y0 = y0 .

(2)

Examples: Consider (X0 , Y0 ) = (0, y0 ) anddXt = −

Yt2
dt + Yt dWt ,2Blanka HorvathdYt = (a + bYt )dt + cdWtH .

(3)

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Our framework and its scope
ExamplesDensity asymptotics

Our setting:dXt = b1 (κ1 , Yt )dt + β σ1 (Yt )dWt ,X0 = x0

dYt = b2 (κ2 , Yt )dt + β σ2 dWtH ,Y0 = y0 .

(2)

Examples: Consider (X0 , Y0 ) = (0, y0 ) anddXt = −

Yt2
dt + Yt dWt ,2

dYt = (a + bYt )dt + cdWtH .

(3)• Define (Xt , Yt ) := (2H−1 X2 t , Y2 t ) ⇒ (2) with κ1 = 2H + 1, κ2 = 2 andβ = 2H, (x0 , y0 ) = (0, y0 ):dXt = −2H+1(Yt )2
dt + 2H Yt dWt ,2

dYt = 2 (a + bYt )dt + 2H cdWtH .
(4)Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Our framework and its scope
ExamplesDensity asymptotics

Our setting:dXt = b1 (κ1 , Yt )dt + β σ1 (Yt )dWt ,X0 = x0

dYt = b2 (κ2 , Yt )dt + β σ2 dWtH ,Y0 = y0 .

(2)

Examples: Consider (X0 , Y0 ) = (0, y0 ) anddXt = −

Yt2
dt + Yt dWt ,2

dYt = (a + bYt )dt + cdWtH .

(3)• Define (Xt , Yt ) := (2H−1 X2 t , Y2 t ) ⇒ (2) with κ1 = 2H + 1, κ2 = 2 andβ = 2H, (x0 , y0 ) = (0, y0 ):dXt = −2H+1(Yt )2
dt + 2H Yt dWt ,2

dYt = 2 (a + bYt )dt + 2H cdWtH .
(4)• Define (Xt , Yt ) := (2H Xt , H Yt ) ⇒ (2) with κ1 = 0, κ2 = β = H,

(x0 , y0 ) = (0, H y0 ):dXt = −(Yt )2
dt + H Yt dWt ,2Blanka HorvathdYt = (aH + bYt )dt + H cdWtH .

(5)

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Our framework and its scope
ExamplesDensity asymptotics

Our setting:dXt = b1 (κ1 , Yt )dt + β σ1 (Yt )dWt ,X0 = x0

dYt = b2 (κ2 , Yt )dt + β σ2 dWtH ,Y0 = y0 .

(2)

Examples: Consider (X0 , Y0 ) = (0, y0 ) anddXt = −

Yt2
dt + Yt dWt ,2

dYt = (a + bYt )dt + cdWtH .

(3)• Define (Xt , Yt ) := (2H−1 X2 t , Y2 t ) ⇒ (2) with κ1 = 2H + 1, κ2 = 2 andβ = 2H, (x0 , y0 ) = (0, y0 ):dXt = −2H+1(Yt )2
dt + 2H Yt dWt ,2

dYt = 2 (a + bYt )dt + 2H cdWtH .
(4)• Define (Xt , Yt ) := (2H Xt , H Yt ) ⇒ (2) with κ1 = 0, κ2 = β = H,

(x0 , y0 ) = (0, H y0 ):dXt = −(Yt )2
dt + H Yt dWt ,2

dYt = (aH + bYt )dt + H cdWtH .

(5)

Remark: Scaling (4) ⇒ short-time (cf. Forde-Zhang (2015)), scaling (5) ⇒ tails.
Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Our framework and its scope
ExamplesDensity asymptotics

Theorem (Harms-H-Jacquier)Consider an SDE of the form (2). Then the density of XTε admits an expansionfε (T , x) = expbTΛ0 (x)XΛ(x)− 2β +
ε
εβ

where


Λ(x) = inf!


ε− min(κ1 ,β) c0 + O(εδ(κ1 ,β) ) ,1
kkk2HH , k ∈ Kxx0 ,y02=

as ε → 0,1
kk0 k2HH ,
2

and


h




i
bt = ∂x b 0, φk0 + ∂x σ φh0 · k̇0 (t) X
bt dt+∂ β b 0, φk0 dt,
dX
t
t
t
ε

b0 = ∂ β x ε |X0 ε=0 ,
ε

where φk0 denotes the ODE solution of the same SDE (2) replacing εβ dW by k̇0 andx0ε by x0 .

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Short-time expansion
Tail expansion
Implied volatility asympotics
Outlook: Refined expansions and moderate regimes

Corollary: Varadhan-type asymptotics
Corollary (short-time asymptotics in Stein-Stein)dYt = (a + bYt )dt + cdWtH

Consider the Stein-Stein model (Xt , Yt ) as in (3) with X0 = 0, Y0 = y0 > 0. Then in aneighbourhood of (x0 , y0 ) the density of Xt satisfies the following asymptotic expansion
as t → 0




Λ(x)
1
−H
δ(H,H+1/2)
fX (t, x) = exp
t
+
O(t
)
t 2H
2π
where


Λ(x) = inf

1
kkk2HH , k ∈ Kxx0 ,y02

Blanka Horvath.

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Short-time expansion
Tail expansion
Implied volatility asympotics
Outlook: Refined expansions and moderate regimes

Corollary: Varadhan-type asymptotics
Corollary (short-time asymptotics in Stein-Stein)dYt = (a + bYt )dt + cdWtH

Consider the Stein-Stein model (Xt , Yt ) as in (3) with X0 = 0, Y0 = y0 > 0. Then in aneighbourhood of (x0 , y0 ) the density of Xt satisfies the following asymptotic expansion
as t → 0




Λ(x)
1
−H
δ(H,H+1/2)
fX (t, x) = exp
t
+
O(t
)
t 2H
2π
where


Λ(x) = inf

1
kkk2HH , k ∈ Kxx0 ,y02.

Proof: Take T = 1, ε2 = t and consider (Xt , Yt ) := (2H−1 X2 t , Y2 t ) with X0 = 0,Y0 = y0 > 0. ⇒ Short-time scaling:dXt = −2H+1(Yt )2
dt + 2H Yt dWt ,2

dYt = 2 (a + bYt )dt + 2H cdWtH ,(4)

Note that the drift vanishes in the limit  → 0 and x0ε = x0 = 0.
bt , Ybt ) ≡ 0, so that there is no 1/β = 1/t β/2 term in the exponential.
⇒ (XBlanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Short-time expansion
Tail expansion
Implied volatility asympotics
Outlook: Refined expansions and moderate regimes

Corollary: tail asymptotics
Corollary (tail expansion in Stein-Stein)dYt = (a + bYt )dt + cdWtH

Consider the Stein-Stein model (3) with X0 = 0, Y0 = y0 > 0. Then as x → ∞,fX (t, x) = exp −c1 x + c2 x 1/2

1
x 1/2




c0 + O x 1/2

bt Λ0 (1).
where c1 := Λ(1), c2 := X
Note that the expression on the RHS is independent of the Hurst-parameter!

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Short-time expansion
Tail expansion
Implied volatility asympotics
Outlook: Refined expansions and moderate regimes

Corollary: tail asymptotics
Corollary (tail expansion in Stein-Stein)dYt = (a + bYt )dt + cdWtH

Consider the Stein-Stein model (3) with X0 = 0, Y0 = y0 > 0. Then as x → ∞,fX (t, x) = exp −c1 x + c2 x 1/2

1
x 1/2




c0 + O x 1/2

bt Λ0 (1).
where c1 := Λ(1), c2 := X
Note that the expression on the RHS is independent of the Hurst-parameter!
Proof: Consider (XT , YT ) := (2H XT , H Yt ) with X0 = 2H X0 and Y0 = H Y0 .
dXt = −(Yt )2
dt + H Yt dWt ,2

dYt = (aH + bYt )dt + H cdWtH ,(5)∆

Note that Xtε = ε2H Xt . ⇒ P(Xtε ≥ y ) = P(Xt ≥ y /ε2H ), ⇒fX (t, y /ε2H ) = ε2H fε (t, y ). Take y = 1, that is x := ε−2H . By the theorem,
Λ(1)fε (t, 1) ≈ exp − ε2 H + . . . ε1H , henceΛ(1)fX (t, x) ≈ exp − ε2H + . . . εH = exp (−Λ(1)x + . . .)Blanka Horvath

1
.
x 1/2

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Short-time expansion
Tail expansion
Implied volatility asympotics
Outlook: Refined expansions and moderate regimes

From density to implied volatility: small-time
Recall the Black-Scholes density expansion:fBS (t, x) ∼ t −1/2 exp

−

1
2t



x − x0σ

2 !
,as t → 0, for any x ∈ R.

(We normalise the spot here, so that x0 = 0).
Our theorem (corollary) says that in the Stein-Stein model (3), we have
d 2 (x0 , y0 ; x)fX (t, x) ∼ cst t −H exp −,2H
2t

Blanka Horvath

as t → 0.

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Short-time expansion
Tail expansion
Implied volatility asympotics
Outlook: Refined expansions and moderate regimes

From density to implied volatility: small-time
Recall the Black-Scholes density expansion:fBS (t, x) ∼ t −1/2 exp

−

1
2t



x − x0σ

2 !
,as t → 0, for any x ∈ R.

(We normalise the spot here, so that x0 = 0).
Our theorem (corollary) says that in the Stein-Stein model (3), we have
d 2 (x0 , y0 ; x)fX (t, x) ∼ cst t −H exp −,2H2t

as t → 0.

Matching the leading-orders gives
σBS (t, x) ∼|x|t H−1/2d(x0 , y0 ; x)Blanka Horvath

as t → 0.

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Short-time expansion
Tail expansion
Implied volatility asympotics
Outlook: Refined expansions and moderate regimes

From density to implied volatility: tails

Recall the Black-Scholes density expansion:x2xfBS (t, x) ∼ exp − 2 −2σ t
4

as x → ∞, for any t > 0.

Our theorem (corollary) says that in the Stein-Stein model (3), we havefX (t, x) ∼√ 
cst
exp −c1 x + c2 x ,x 1/2

Blanka Horvath

as x → ∞.

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Short-time expansion
Tail expansion
Implied volatility asympotics
Outlook: Refined expansions and moderate regimes

From density to implied volatility: tails

Recall the Black-Scholes density expansion:x2xfBS (t, x) ∼ exp − 2 −2σ t
4

as x → ∞, for any t > 0.

Our theorem (corollary) says that in the Stein-Stein model (3), we havefX (t, x) ∼√ 
cst
exp −c1 x + c2 x ,x 1/2

as x → ∞.

Matching the leading-orders gives√x
x2−c1 x + c2 x ∼ − 2 − ,2σ t
4
and we recover Roger Lee’s formula independently of the Hurst exponent in (3).

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Short-time expansion
Tail expansion
Implied volatility asympotics
Outlook: Refined expansions and moderate regimes

Outlook: Moderate regimes• Moderate Regimes (in the sense of Friz-Gerhold-Pinter ’16) interpolate between
out-of-the-money calls with fixed strike log SK = k > 0 and at-the-money0

k = 0 calls: Now kt = ct θ ⇒ MOTM (for 0 < θ < 12 ) and AATM (for larger θ)• Reflects market data: options closer expiry ⇒ strikes closer to the money

first observed by Mijatović-Tankov on FX markets• The moderate regime (MOTM) permits explicit computations for the ratefunction Λ(k) in terms of the model parameters
Moderate deviations⇒ Advantage over OTM (large deviations) case where theΛ(k) often related to geodesic distance problems and not explicitly available.
• MOTM expansions naturally involve quantities very familiar to practitioners,

notably spot (implied) volatility, implied volatility skew . . .
• In some cases (fractional volatility models) the scaling θ permits a fine-tuning tounderstand the behavior and derivatives of the energy function.

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Short-time expansion
Tail expansion
Implied volatility asympotics
Outlook: Refined expansions and moderate regimes

Moderate regimes for rough volatilityRescalings =⇒ We tacitly agreed to consider P Xt ≈ t 1/2−H x . Now it is only asmall step to consider instead (for some suitable small θ > 0)
P Xt ≈ t 1/2−H+θ x .

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Short-time expansion
Tail expansion
Implied volatility asympotics
Outlook: Refined expansions and moderate regimes

Moderate regimes for rough volatilityRescalings =⇒ We tacitly agreed to consider P Xt ≈ t 1/2−H x . Now it is only asmall step to consider instead (for some suitable small θ > 0)
P Xt ≈ t 1/2−H+θ x .
Theorem (Bayer-Friz-Gulisashvili-H-Stemper)
Consider a moderately out-of-the-money call kt = x 1/2−H+θ ; θ ∈ (0, H) resp.
θ ∈ (0, 2H). Then as t → 0, the following (non-Markovian extention of Osajima3energy-expansion) holds
log c(kt , t) ≈

1 00
x2
1
x3
Λ (0) 2H−2θ + Λ000 (0) 2H−3θ ,
2
t
6
t
6σ 0

where we have explicit expressions: Λ00 (0) = σ1 ) and Λ000 (0) = −ρ σ40 hK , 1i.
00
R RHere K denotes the Volterra kernel and hK , 1i := 01 0t K (t, s)dsdt.

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Notations
Sketch of the proof

Notations• H: absolutely continuous paths [0, T ] → R2 starting at 0 such that

2

ḣ

H

< ∞.

• HH := KH H and k := KH h, where KH denotes the Volterra kernel.
• For fixed (x0 , y0 ) ∈ R2 , φk is the (unique) ODE solution to
m
 
 
X
i
φ̇kt = σ0 φkt dt +σ i φht dkt ,

φk0 = (x0 , y0 ).

i=1• Denote ψ k := Π1 φk its projection on to the first coordinate X .

k = a ∈ R 6= ∅ (”by Hörmander condition”).
• Ka := k ∈ HH : ψT
• Λ(a) := infn1
kkk2H : k ∈ Ka
2o.

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Notations
Sketch of the proof

Proof of the theorem 1

1
dXt = −2H+1 Yt2 dt + 2H Yt dWt ,2

dYt = 2H dWtH ,with the same initial condition X0 = Y0 = 0.
#
"


bT
Λ0 (x)X
Λ(x)ε−2H c0 + O(ε2H ) .
Density: fε (T , x) = exp − 4H +2HεεBlanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Notations
Sketch of the proof

Proof of the theorem 1

1
dXt = −2H+1 Yt2 dt + 2H Yt dWt ,2

dYt = 2H dWtH ,with the same initial condition X0 = Y0 = 0.
#
"


bT
Λ0 (x)X
Λ(x)ε−2H c0 + O(ε2H ) .
Density: fε (T , x) = exp − 4H +2Hεε
Proof: Take x ∈ R and a C ∞ -bounded function F such that F (x) = 0.


 ε


Z
XT − (x, 0)
F (XTε )4H1fε (T , x)e−F (x)/ε =
−
dζ.
E
exp
i(ζ,
0)
·
2πε2H R
ε2H
ε4H
Choose F such that F (·) + Λx0 (·) has a non-degenerate minimum at z. This implies
that k 7→ F (φkT (x0 , y0 )) + 21 kkk2HH has a non-degenerate minimum at k0 ∈ H.
(For instance F (z) = λ|z − x|2 − [Λx0 ,y0 (z) − Λx0 ,y0 (x)] with λ > 0).

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Notations
Sketch of the proof

Proof of the theorem 2
Replace ε2H dB (B := (W , W H )) in the SDE by ε2H dW + k̇0 .
e ε ):eε, YCall the corresponding Girsanov-transformed process Zetε = (Xtte ε = −ε2H+1 1 Y
e 2 dt + Y
e ε (ε2H dWt + (k̇0 )1 ),dX
2

e = 2H dW H + (k̇0 )2 .
dYt

Girsanov factor


Z T
1
1
ψ(k0 )t dBt − 4H kk0 k2HH .
G = exp − 2H20Therefore
4H

f (x, T )e −F (x)/4εZ
h 2H e
i
−4H
1
eT )
F (X
E e  iζ(XT −x)−ε
G dζ
2H
2π
ZR h
i
1
(∗)
=
E
e
dζ
2πε2H R=

whereeT − x) − −4H F (XeT ) − −2H(∗) = ε2H iζ(XZ T
0Blanka Horvath

1
ψ(γ)t dBt − −4H kγk21/2,H .
2

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Notations
Sketch of the proof

Proof of the theorem 2
Replace ε2H dB (B := (W , W H )) in the SDE by ε2H dW + k̇0 .
e ε ):eε, YCall the corresponding Girsanov-transformed process Zetε = (Xtte ε = −ε2H+1 1 Y
e 2 dt + Y
e ε (ε2H dWt + (k̇0 )1 ),dX
2

e = 2H dW H + (k̇0 )2 .
dYt

Girsanov factor


Z T
1
1
ψ(k0 )t dBt − 4H kk0 k2HH .
G = exp − 2H20e ε ) for ε2H → 0,eε, YBy a stochastic Taylor expansion of Zetε = (X
t
t
 




Z T
eε
−F X
t
2H b
0
4H
 = exp −1 F (x) − ε2H
exp 
ψ(k
)
dB
−
ε
X
·
Λ
(x)
+
O(ε
)
t
t
0
T
x
0
ε4H
ε4H
0

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Notations
Sketch of the proof

Proof of the theorem 2
Replace ε2H dB (B := (W , W H )) in the SDE by ε2H dW + k̇0 .
e ε ):eε, YCall the corresponding Girsanov-transformed process Zetε = (Xtte ε = −ε2H+1 1 Y
e 2 dt + Y
e ε (ε2H dWt + (k̇0 )1 ),dX
2

e = 2H dW H + (k̇0 )2 .
dYt

Girsanov factor


Z T
1
1
ψ(k0 )t dBt − 4H kk0 k2HH .
G = exp − 2H20e ε ) for ε2H → 0,eε, YBy a stochastic Taylor expansion of Zetε = (X
t
t
 



Z T
eε
−F X
t
1 b
1
0
 = exp
exp 
ψ(k
)
dB
+
X
·
Λ
(x)
+
O(1)
.
t
t
0
T
x
0
ε4H
ε2H 0
ε2H
The rest of the proof follows Ben Arous’ proof for XTε .

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models

Introduction
Main result and motivation
Corollaries and outlook
Proof

Notations
Sketch of the proof

Thank you!

Blanka Horvath

Asymptotic expansions for fractional stochastic volatility models